Question $\frac{-4 x+1}{5 x+4}>-\frac{3}{2}$

Studdy Solution

STEP 1

What is this asking? We need to figure out when the fraction $\frac{-4x + 1}{5x + 4}$ is greater than $-\frac{3}{2}$ . Watch out! Be careful with the inequality direction when multiplying or dividing by negative numbers!

STEP 2

1. Move everything to one side
2. Find a common denominator
3. Solve the inequality

STEP 3

Alright, let's start by getting everything on one side of the inequality.
We have:
$\frac{-4x + 1}{5x + 4} > -\frac{3}{2}$
To do this, we'll subtract $-\frac{3}{2}$ from both sides:
$\frac{-4x + 1}{5x + 4} + \frac{3}{2} > 0$

STEP 4

Now, we need a common denominator to combine these fractions.
The denominators are $5x + 4$ and $2$ .
So, our common denominator will be $2(5x + 4)$ .

STEP 5

Let's rewrite each fraction with this common denominator:
$\frac{-4x + 1}{5x + 4} = \frac{-4x + 1}{5x + 4} \cdot \frac{2}{2} = \frac{2(-4x + 1)}{2(5x + 4)}$
$-\frac{3}{2} = -\frac{3}{2} \cdot \frac{5x + 4}{5x + 4} = \frac{-3(5x + 4)}{2(5x + 4)}$

STEP 6

Combine the fractions:
$\frac{2(-4x + 1) + (-3)(5x + 4)}{2(5x + 4)} > 0$

STEP 7

Simplify the numerator:
$2(-4x + 1) = -8x + 2$
$-3(5x + 4) = -15x - 12$

STEP 8

Combine these results:
$-8x + 2 - 15x - 12 = -23x - 10$

STEP 9

Now, our inequality is:
$\frac{-23x - 10}{2(5x + 4)} > 0$

STEP 10

The fraction is greater than zero when the numerator and denominator have the same sign.
So, let's solve:
1. Numerator positive: $-23x - 10 > 0$
$ -23x > 10 \quad \Rightarrow \quad x < -\frac{10}{23} \]
2. Denominator positive: $5x + 4 > 0$
$ 5x > -4 \quad \Rightarrow \quad x > -\frac{4}{5} \]

STEP 11

Combine these intervals.
The solution is:
$-\frac{4}{5} < x < -\frac{10}{23}$

STEP 12

The values of $x$ that satisfy the inequality are:
$-\frac{4}{5} < x < -\frac{10}{23}$

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